scholarly journals Local Dynamics of Logistic Equation with Delay and Diffusion

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1566
Author(s):  
Sergey Kashchenko
Keyword(s):  
Boundary Value Problem ◽  
Equilibrium State ◽  
Boundary Value ◽  
Small Neighborhood ◽  
Logistic Equation ◽  
Asymptotic Formulas ◽  
Parametric Perturbation ◽  
Original Boundary ◽  
And Diffusion ◽  
Equation With Delay

The behavior of all the solutions of the logistic equation with delay and diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small perturbations of all coefficients are considered, including the delay coefficient and the coefficients of the boundary conditions. The conditions are studied when these perturbations depend on the spatial variable and when they are time-periodic functions. Equations on the central manifold are constructed as the main results. Their nonlocal dynamics determines the behavior of all the solutions of the original boundary value problem in a sufficiently small neighborhood of the equilibrium state. The ability to control the dynamics of the original problem using the phase change in the perturbing force is set. The numerical and analytical results regarding the dynamics of the system with parametric perturbation are obtained. The asymptotic formulas for the solutions of the original boundary value problem are given.

scholarly journals Finite-dimensional maps describing the dynamics of a logistic equation with delay

2019 ◽  
Vol 487 (6) ◽  
pp. 611-616
Author(s):  
S. D. Glyzin ◽  
S. A. Kashchenko
Keyword(s):  
Numerical Simulation ◽  
Equilibrium State ◽  
Stable Equilibrium ◽  
Dynamic Properties ◽  
Logistic Equation ◽  
Mathematical Ecology ◽  
Stable Equilibrium State ◽  
Finite Dimensional ◽  
Equation With Delay ◽  
Dynamics Of Populations

This article discusses a family of maps that are used in the numerical simulation of a logistic equation with delay. This equation and presented maps are widely used in problems of mathematical ecology as models of the dynamics of populations. The paper compares the dynamic properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of maps can be quite complicated, while the logistic equation with delay has only a stable equilibrium state or cycle.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

scholarly journals Asymptotic Solution of a Boundary Value Problem for a Spring–Mass Model of Legged Locomotion

2020 ◽  
Vol 30 (6) ◽  
pp. 2971-2988
Author(s):  
Hanna Okrasińska-Płociniczak ◽  
Łukasz Płociniczak
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Asymptotic Expansions ◽  
Boundary Value ◽  
Legged Locomotion ◽  
Asymptotic Formulas ◽  
Numerical Verification ◽  
Mass Model ◽  
Leg Stiffness ◽  
Elastic Pendulum

Abstract Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions of the stiffness. These are conducted perturbatively: once with respect to small angles of attack, and once for large velocities. Our findings are in agreement with previous results and numerical simulations. In particular, we show that the leg stiffness is inversely proportional to the square of the attack angle for its small values, and proportional to the velocity for large speeds. We give exact asymptotic formulas to several orders and conclude the paper with a numerical verification.

NONLINEAR STABILITY OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE BROADWELL MODEL AROUND A MAXWELLIAN

2011 ◽  
Vol 08 (01) ◽  
pp. 131-157 ◽  
Author(s):  
HUEY-ER LIN
Keyword(s):  
Boundary Value Problem ◽  
Equilibrium State ◽  
Small Perturbation ◽  
Nonlinear Stability ◽  
Pointwise Convergence ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Solution Representation ◽  
Initial Boundary Value

We study the nonlinear stability of the equilibrium state associated with the initial boundary value problem for the Broadwell model with transonic boundary. Based on the Green's function of this linearized initial boundary value problem and on the understanding of its action on microscopic data, we establish the pointwise convergence of the solution toward the equilibrium state under small perturbation, via a solution representation and Picard-like iterations.

scholarly journals Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Rauf Kh. Amırov ◽  
A. Adiloglu Nabıev
Keyword(s):  
Inverse Problems ◽  
Boundary Value Problem ◽  
Finite Interval ◽  
Boundary Value ◽  
Weyl Function ◽  
Integral Representations ◽  
Asymptotic Formulas ◽  
Linearly Independent ◽  
Quadratic Pencil ◽  
Sturm Liouville

In this study some inverse problems for a boundary value problem generated with a quadratic pencil of Sturm-Liouville equations with impulse on a finite interval are considered. Some useful integral representations for the linearly independent solutions of a quadratic pencil of Sturm-Liouville equation have been derived and using these, important spectral properties of the boundary value problem are investigated; the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers are obtained. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.

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